Abstract
The Bethe equations which arise in description of the spectrum of the dilatation operator for the su(2) sector of the \( \mathcal{N} = 4 \) supersymmetric Yang-Mills theory are considered in the anti-ferromagnetic regime. These equations are a deformation of those for the Heisenberg XXX magnet. We prove that in the thermodynamic limit, roots of the deformed equations group into strings. We prove that the corresponding Yang's action is convex, which implies the uniqueness of a solution for centers of the strings. The state formed by strings of length (2n+1) is considered, and the density of their distribution is found. It is shown that the energy of such a state decreases as n grows. It is observed that the nonanalyticity of the left-hand sides of the Bethe equations leads to an additional contribution to the density and energy of strings of even length. We conclude that the structure of the anti-ferromagnetic vacuum is determined by the behavior of exponential corrections to string solutions in the thermodynamic limit and, possibly, involves strings of length 2. Bibliography: 14 titles.
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References
L. N. Lipatov, “High energy asymptotics of multi-color QCD and exactly solvable lattice models;” hep-th/9311037.
L. D. Faddeev and G. P. Korchemsky, “High energy QCD as a completely integrable model,” Phys. Lett., B342, 311–322 (1995).
V. M. Braun, S. E. Derkachov, and A. N. Manashov, “Integrability of three particle evolution equations in QCD,” Phys. Rev. Lett., 81, 2020–2023 (1998).
J. A. Minahan and K. Zarembo, “The Bethe ansatz for \( \mathcal{N} = 4 \) super Yang-Mills,” JHEP, 0303, 013 (2003).
L. D. Faddeev and L. A. Takhtajan, “Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model,” J. Sov. Math., 24, 241–267 (1984).
L. D. Faddeev, “How algebraic Bethe ansatz works for integrable model,” in: Symmétries Quantiques, Les Houches, 1995 (North-Holland, Amsterdam, 1998), pp. 149–219; hep-th/9605187.
V. Korepin, N. Bogoliubov, and A. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Univ. Press (1993).
N. Beisert, V. Dippel, and M. Staudacher, “A novel long range spin chain and planar \( \mathcal{N} = 4 \) super Yang-Mills,” JHEP, 0407, 075 (2004).
K. Zarembo, “Antiferromagnetic operators in \( \mathcal{N} = 4 \) supersymmetric Yang-Mills theory,” Phys. Lett., B634, 552–556 (2006).
A. Rej, D. Serban, and M. Staudacher, “Planar \( \mathcal{N} = 4 \) gauge theory and the Hubbard model,” JHEP, 0603, 018 (2006).
L. A. Takhtajan, “The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins,” Phys. Lett., A87, 479–482 (1981/82).
H. M. Babujian, “Exact solution on the one-dimensional isotropic Heisenberg chain with arbitrary spin S,” Phys. Lett., A90, 479–482 (1982).
H. J. de Vega, L. Mezincescu, and R. I. Nepomechie, “Thermodynamics of integrable chains with alternating spins,” Phys. Rev., B49, 13223–13226 (1994).
A. G. Bytsko and A. Doikou, “Thermodynamics and conformal properties of XXZ chains with alternating spins,” J. Phys., A37, 4465–4492 (2004).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 75–87.
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Bytsko, A.G., Shenderovich, I.E. On string solutions of the Bethe equations in the \( \mathcal{N} = 4 \) supersymmetric Yang-Mills theory. J Math Sci 151, 2840–2847 (2008). https://doi.org/10.1007/s10958-008-9004-8
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DOI: https://doi.org/10.1007/s10958-008-9004-8